In this paper, we price a compound option with log asset price following an extended variance gamma process. The extended variance gamma process can control the skewness and kurtosis. The parameters of the model are estimated via the maximum likelihood method from historical data. We start with finding the risk neutral Esscher measure under which the discounted asset price process is a martingale. Then we derive an analytical pricing formula for compound option in terms of the Fourier integral of the characteristic function of extended variance gamma process, and we use this formula, in combination with the FFT algorithm, to calculate the compound option price across the whole spectrum of the exercise price. Finally, we present some numerical results for illustration.

In this paper, we price the vulnerable option under the assumption that both underlying and counterparty asset price follow correlated jump diffusion processes. By the two-dimensional Itô-Doeblin formula, we initially derive a partial integro-differential equation (PIDE) satisfied by the price of the vulnerable option with flexible jumps. Then, the double Mellin transform converts the PIDE into an ordinary differential equation (ODE), which possesses an explicit solution. Further, the inverse Mellin transform of the explicit solution to the ODE turns out to be a rigorous solution to the PIDE for vulnerable option. Finally, by carrying out some numerical experiments, we demonstrate that the PIDE’s solution produces price with a convincing degree of precision.

A system of mutually interacting superprocesses with migration is constructed as the limit of a sequence of branching particle systems arising from population models. The uniqueness in law of the superprocesses is established using the pathwise uniqueness of a system of stochastic partial differential equations, which is satisfied by the corresponding system of distribution function-valued processes.